We study here instability problems of standing waves for the nonlinear Klein-Gordon equations and solitary waves for the generalized Boussinesq equations. It is shown that those special wave solutions may be strongly unstable by blowup in finite time, depending on the range of the wave’s frequency or the wave’s speed of propagation and on the nonlinearity.

In this paper, the Exp-function method is used to obtain generalized travelling wave solutions with free parameters of the MKdV-sine-Gordon and Boussinesq-double sine-Gordon equations. It is shown that the Exp-function method, with the help of any symbolic computation packages, provides an effective mathematical tool for nonlinear evolution equations arising in mathematical physics.

Nonlinear coupling among wave modes and vortical modes is investigated with the following question in mind: Can we distinguish the wave-vortical interactions largely responsible for formation versus evolution of coherent, balanced structures? The two main case studies use initial conditions that project only onto the vortical-mode flow component of the rotating Boussinesq equations: (i) an init...

in recent years, the number of research works devoted to applying the highly accurate numerical schemes, in particular compact finite difference schemes, to numerical simulation of complex flow fields with multi-scale structures, is increasing. the use of compact finite-difference schemes are the simple and powerful ways to reach the objectives of high accuracy and low computational cost. compa...

In this paper, we propose the extended Boussinesq–Whitham–Broer–Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider BWBK equations, Whitham–Broer–Kaup (WBK) Boussinesq by setting proper smooth functions that are derived from proposed equation. obtain uniformly coupled traveling wave solutions of considered employing improved system method, subsequently their ...

Standard perturbation methods are applied to Euler’s equations of motion governing the capillary-gravity shallow water waves to derive a general higher-order Boussinesq equation involving the small-amplitude parameter, α = a/h0, and long-wavelength parameter, β = (h0/l), where a and l are the actual amplitude and wavelength of the surface wave, and h0 is the height of the undisturbed water surf...

Parts I and II of this paper describe the extension and testing of two sets of Boussinesq-type equations to include surf zone phenomena. Part I is restricted to 1D tests of shoaling, breaking, and runup, while Part II deals with two horizontal dimensions. The model uses two main extensions to the Boussinesq equations: a momentum-conserving eddy viscosity technique to model breaking, and a ‘‘slo...

In inviscid fluid flows instability arises generically due to a resonance between two wave modes. Here, it is shown that the structure of the weakly nonlinear régime depends crucially on whether the modal structure coincides, or remains distinct, at the resonance point where the wave phase speeds coincide. Then in the weakly nonlinear, long-wave limit the generic model consists either of a Bous...

In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: utt − Luxx = B(±|u|u)xx, p > 1. The main characteristic of this class of equations is the existence of two sources of dispersion, characterized by two coercive pseudo-differential operatorsL and B. Members of the class arise as mathematical models ...